Theorem nnssn0s 28492

Description: The positive surreal integers are a subset of the non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)

Assertion

Ref	Expression
nnssn0s	⊢ ℕs ⊆ ℕ0s

Proof of Theorem nnssn0s

Step	Hyp	Ref	Expression
1		df-nns 28486	. 2 ⊢ ℕs = (ℕ0s ∖ { 0s })
2		difss 4091	. 2 ⊢ (ℕ0s ∖ { 0s }) ⊆ ℕ0s
3	1, 2	eqsstri 3984	1 ⊢ ℕs ⊆ ℕ0s

Syntax hints:   ∖ cdif 3903   ⊆ wss 3906  {csn 4590   0_s c0s 27976  ℕ_0scn0s 28483  ℕ_scnns 28484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-dif 3909  df-ss 3923  df-nns 28486

This theorem is referenced by:  nnssno  28493  nnn0s  28498  nnn0sd  28499  nnsgt0  28510